On interpolation and extrapolation theorem in Orlicz spaces
Project/Area Number  09640206 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
解析学

Research Institution  Oita University 
Principal Investigator 
HIROO Kita Fuculty of Education Professor, 教育学部, 教授 (20224941)

CoInvestigator(Kenkyūbuntansha) 
TAKEMOTO Yoshio Nihon Bunri Univ.Fuculty of Engineering Professor, 工学部, 教授 (20140965)
KEMOTO Nobuyuki Oita University Fuculty of Education Asociated Professor, 教育学部, 助教授 (70161825)
MORI Naganori Oita University Fuculty of Education Professor, 教育学部, 教授 (40040737)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥1,300,000 (Direct Cost : ¥1,300,000)
Fiscal Year 1998 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1997 : ¥800,000 (Direct Cost : ¥800,000)

Keywords  Fourier series / Orlicz space / a.e.convergence / maximal function / フーリエ級数 / オーリッツ空間 / 概収束 / 最大値関数 / オリッツ空間 / フーリェ級数 
Research Abstract 
The first research product of our study in 1998 is as follows. Let *(t) be an increasing function defined on the interval [O,*) satisfying *(0) 0. Let L*(T) be an Orlicz space on T = ["p, pi}. We denote by S_n(f, x) the nth partial sum of the Fourier series of an integrable function f. And S^*(f) means the Fourier maximal operator. When *(t) exp(V^<gamma>)  1, (gamma > 0), we denote by L(expf^<gamma>) the Orlicz space generated by this function *(t). It was proved that if f is a function in the Orlicz space L(expt^<gamma>), then S^*.(f) is in L(expt^<gamma>/^(gamma^<c+>^1^)). This result was already shown in our paper in detail (Acta Math. Hungar. 1994). A generalization of the result mentioned above can be considered. In our previous paper, Young function * was restricted. However in our recent paper this restriction was removed. Our main idea of the proof of this result is an interpolation theory of quasi linear operators in Lorentz spaces. Let *(t) be a rapid]y increasing Young function and L*(T) be an Orlicz space defined by tbis *(t). We could find the sharp Young function * such that S^*(f) is in L*(T) for all f in L*(T). The second research product of our study is a result of almost everywhere convergence of Fourier series of functions in an Orhicz space near to Zygmund class L log L.We have the following result. If l^t_1(*(u))/du<greater than or equal>a_0 log(1+jogt) for t>1 holds, then we get *(S^*(f)LA^1 for all f * L* (see H.Rita [Kil]).

Report
(4results)
Research Output
(12results)